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Citation: Lerbet, Jean, Aldowaji, Marwa, Challamel, Noël, Kirillov, Oleg, Nicot, François and Darve, Félix (2014) Geometric degree of nonconservativity. Mathematics and Mechanics of Complex Systems, 2 (2). pp. 123-139. ISSN 2326-7186

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NISSUNA UMANA INVESTIGAZIONE SI PUO DIMANDARE VERA SCIENZIAS’ESSA NON PASSA PER LE MATEMATICHE DIMOSTRAZIONI

LEONARDO DAVINCI

Mathematics and Mechanicsof

Complex Systems

msp

vol. 2 no. 2 2014

JEAN LERBET, MARWA ALDOWAJI, NOËL CHALLAMEL,OLEG N. KIRILLOV, FRANÇOIS NICOT AND FÉLIX DARVE

GEOMETRIC DEGREE OF NONCONSERVATIVITY

MATHEMATICS AND MECHANICS OF COMPLEX SYSTEMSVol. 2, No. 2, 2014

dx.doi.org/10.2140/memocs.2014.2.123MM ∩

GEOMETRIC DEGREE OF NONCONSERVATIVITY

JEAN LERBET, MARWA ALDOWAJI, NOËL CHALLAMEL,OLEG N. KIRILLOV, FRANÇOIS NICOT AND FÉLIX DARVE

This paper deals with nonconservative mechanical systems subjected to noncon-servative positional forces leading to nonsymmetric tangential stiffness matrices.The geometric degree of nonconservativity of such systems is then defined asthe minimal number ` of kinematic constraints necessary to convert the initialsystem into a conservative one. Finding this number and describing the set ofcorresponding kinematic constraints is reduced to a linear algebra problem. Thisindex ` of nonconservativity is the half of the rank of the skew-symmetric partKa of the stiffness matrix K that is always an even number. The set of constraintsis extracted from the eigenspaces of the symmetric matrix K 2

a . Several examplesincluding the well-known Ziegler column illustrate the results.

Introduction

This paper is concerned with the statics or dynamics of a discrete or discretizedsystem 6free. We assume that, after having started with different possible nonlinearsettings, convenient assumptions and approximations lead to a dynamic evolutiongoverned by the following equation of motion of the free system 6free:

M X + K (p)X = 0, (1)

where K (p) = Ks(p)+ Ka(p), with Ks(p) = 12 (K (p)+ K T (p)) and Ka(p) =

12 (K (p)−K T (p)); note that K (p) is generally a nonsymmetric matrix (Ka(p) 6= 0)because of the nonconservativity of 6free, whereas M is symmetric positive definite.

The paradoxical effects of mechanical systems linearly governed by (1) with anonsymmetric stiffness matrix have been known for some time (see for example[Ziegler 1952; Bolotin 1961]) and have been investigated in depth, especially con-cerning the so-called destabilizing effect of adding friction in the system (see [Kir-illov and Verhulst 2010] for recent developments). Bigoni and Noselli [2011] haveillustrated through an experimental device calculations starting from dynamics withdry friction and coming to equations like (1). Circulatory forces seem to have

Communicated by Francesco dell’Isola.MSC2010: 15A18, 70G99.Keywords: linear algebra, nonconservative system.

123

124 LERBET, ALDOWAJI, CHALLAMEL, KIRILLOV, NICOT AND DARVE

appeared first in the rotor dynamics works of early 1920s. It is generally agreedthat E. L. Nikolai was the first to have found some curious paradoxes inducedby the nonconservative aspect of the loading system. Here, we only investigatesystems without rotating effects which means that circulatory forces are only non-conservative positional loading. More precisely, by decomposing in the linear casethe stiffness matrix K (p) into a symmetric part and a skew-symmetric part, onethen decomposes nonpotential forces into a potential component and a circulatorycomponent. Zhuravlev [2007] suggests an extension of this decomposition for thenonlinear case by using the trick of Poincaré in his theorem about exact and closeddifferential forms. This could be a good way to tackle the nonlinear extension ofthe present paper.

If p is a loading parameter any norm ‖Ka(p)‖ of the skew-symmetric part Ka(p)of K (p) is an elementary measure of the nonconservativity of the correspondingnonpotential forces by any norm on the space of matrices. However, this roughmeasure indicates the amplitude of the nonconservativity and masks another moreintrinsic measure of this nonconservativity which is defined in this paper. This ishere defined by a lower semicontinuous function with only finite integer values (foran increasing load). This function is then locally independent on the load parametervalue, except perhaps for a finite number of singular values {p∗0 < p∗1 < · · ·< p∗r } ofp. Obviously p∗0 = 0 is such a value, because for p = 0 the system is conservativeand Ka(0) = 0. In all the examples except the so-called Bigoni system, the onlyvalue is p = 0. Because it is also linked to a dimension of a linear space, wethen propose to call this number the geometric degree of nonconservativity of thesystem (or of the forces).

The genesis of the used approach lies in several papers [Challamel et al. 2009;2010; Nicot et al. 2011; Lerbet et al. 2012] that investigated the deep rule of thesecond-order work criterion proposed in [Hill 1958] for solids in the frameworkof nonassociated plasticity, and independently also proposed for instabilities forsystems subjected to nonpotential forces in [Absi and Lerbet 2004]. This criterionperforms especially well for nonconservative systems because, contrary to the di-vergence criterion, it remains “stable” under the action of additional kinematicsconstraints: if this criterion holds for a free system and for a value p of the loadparameter, it still holds for the same value p and for a system subjected to anyfamily of additional kinematic constraints. This property, contrary to a similarwell-known consequence of the Rayleigh theorems for conservative systems, isgenerally no more valid for nonconservative systems. This paradoxical behaviorof the mechanical system, or more precisely of the stability of the investigated equi-librium configuration of the mechanical system when adding additional kinematicconstraints, is actually a characteristic of nonconservative systems.

Thus, extending the above-mentioned works concerning the effects of additional

GEOMETRIC DEGREE OF NONCONSERVATIVITY 125

constraints on such n-DOF (degree of freedom) nonconservative mechanical sys-tems [Challamel et al. 2010; Lerbet et al. 2012; 2013], we focus on families ofkinematic constraints that could convert 6free into a conservative system. Moreprecisely, we address both the problems of the existence of a minimal family (ac-cording to the number of constraints) of such constraints and that of building theset of such families. The minimal number of constraints required to convert thenonconservative system 6free into a conservative system is then a measure of thenonconservativity of 6free and will be called the geometric degree of nonconserva-tivity of 6free.

The paper is organized as follows: in Section 1, the mechanical problem isreduced to a linear algebra problem. In Section 2, the solution is developed leadingto the concept of the geometric degree of nonconservativity of a mechanical system.In Section 3, several examples illustrate the mathematical results.

1. Modeling of the mechanical problem

Let 6free be a n-DOF discrete mechanical system and suppose, as above, that thedynamic evolution of 6free is governed by (1). X is the vector of kinematic un-knowns (X T

= (x1, . . . , xn) ∈M1n(R)), M is the mass matrix (symmetric definitepositive), and K = K (p) is the stiffness matrix. The latter is any square matrixbecause of the nonconservativity of 6free. Let p be a (loading) parameter. Supposethat m (independent) additional kinematic constraints C1, . . . ,Cm are set up on6free. The linear framework leads us to model each kinematic constraint C j bya linear relationship

∑nk=1 α

jk xk = 0. Thus C j is represented by and identified

with a vector α j= (α j

1 , . . . , αjn ) of Rn (actually it is a linear form on Rn but

by the canonical scalar product we may identify both spaces). The family of mconstraints {α1, . . . , αm

} may be considered as an element of an nm-dimensionalvector space — for instance as an n×m matrix A =

(α1· · · αm

)in Mnm(R), or

more precisely in Gnm(R), the open subset of matrices of Mnm(R) with rank m,because of the independence of the constraints. If m is fixed (it will have to befound in a first step), we then have to find the set Cm(6free)⊂ Gnm(R) such that ifA ∈ Cm(6free) then the constrained mechanical system 6cons =6cons(A) becomesconservative. Thus

AT=

α1T

...

αmT

=α

11 · · · α

1n

.... . .

...

αm1 · · · α

mn

(every vector αiT

= (αi1, . . . , α

in) could be normalized αiTαi

= 1).Let 3 ∈Mm1(R), with 3T

= (λ1 . . . λm), be the Lagrange multiplier attachedto the constraints. The equation of motion of the constrained system 6cons(A) is

126 LERBET, ALDOWAJI, CHALLAMEL, KIRILLOV, NICOT AND DARVE

AT X = 0, (2)

M X + K (p)X + A3= 0, (3)

Let T (A)=Vect{α1, . . . , αm} and let H(A)= T (A)⊥ be the orthogonal to T (A)

in Rn identified with Mn1(R). Thus dim T (A)= m and dim H(A)= n−m.Let us choose an orthonormal basis of T (A) (by Gram–Schmidt from (α1, . . . ,

αm), for example) (t1(A), . . . , tm(A)) and another (hm+1(A), . . . , hn(A)) of H(A)such that b(A)= (t1(A), . . . , tm(A), hm+1(A), . . . , hn(A)) is an orthonormal basisof Rn and let P = P(A) ∈ On(R) be the orthogonal matrix passing from thecanonical basis of Rn to b(A):

P = P(A)=mat(t1(A), . . . , tm(A), hm+1(A), . . . , hn(A)).

Let Y be defined by X = P(A)Y . The previous system reads:

(P(A)T A)T Y = 0, (4)

PT (A)M P(A)Y + PT (A)K (p)P(A)Y + P(A)T A3= 0, (5)

Considering Mcons(A) (resp. Kcons(A, p)) the square submatrix of PT (A)M P(A)(resp. PT (A)K (p)P(A)) built by suppressing the first m rows and the first mcolumns of PT (A)M P(A) (resp. of PT (A)K (p)P(A)), we get the following equa-tions of the constrained system without the Lagrange multipliers:

Mcons(A)Ycons+ Kcons(A, p)Ycons = 0, (6)

where Y Tcons = (ym+1, . . . , yn) ∈M1 n−m(R).

We are then led to investigate when Kcons(A, p), a (n−m) × (n−m) squaresubmatrix of PT (A)K (p)P(A), is symmetric. Note that, in the standard case ofstructural mechanics, K (p)= Kel− pKext with Kel the symmetric definite-positivestiffness matrix relative to elastic actions and Kext the nonsymmetric matrix relativeto external actions (circulatory force). Kcons(A, p) reads:

Kcons(A, p)=

hTm+1(A)K (p)hm+1(A) · · · hT

m+1(A)K (p)hn(A)...

. . ....

hTn (A)K (p)hm+1(A) · · · hT

n (A)K (p)hn(A)

.The condition for the constraints defined by A to convert the free nonconserva-

tive system into a conservative one is then

hTi (A)K (p)h j (A)= hT

i (A)KT (p)h j (A) for all i, j,

or, in a more geometrical phrasing: for every pair u and v of two orthogonal vectorsof H(A) orthogonal to T (A) = Vect{α1, . . . , αm

}, uT K (p)v = uT K T (p)v. This

GEOMETRIC DEGREE OF NONCONSERVATIVITY 127

is equivalent to uT Ka(p)v = 0 for every pair u and v of any two vectors of H(A)with Ka(p) the skew-symmetric part of K (p). This is obviously right for m = n−1because for any n− 1 independent constraints the submatrix becomes a scalar, theskew-symmetric part of which is always nil!

Let φa(p) be the linear map whose matrix is in the canonical basis of Rn isKa(p). Geometrically, the condition means that for every vector u of H(A),φa(p)(u) is orthogonal to H(A) or equivalently belongs to T (A):

φa(p)(u) ∈ T (A) for all u ∈ H(A). (7)

The initial mechanical problem has then been modeled into the following originalproblem of linear algebra and more precisely of Euclidean spaces: Does there existan (n−m)-dimensional subspace of Rn which is sent onto its orthogonal by φa(p)?We denote by ( · | · ) the scalar product of Rn ((u | v)= uT v with the identificationof Rn with Mn1(R)). In the following section, this problem is solved.

2. Solution of the mathematical modeling

We forget the p-dependency of all the quantities. In the introduction, we alreadynoted that the loading interval I = [0,+∞[ may be decomposed as {0} ∪ ]0, p∗1] ∪]p∗1, p∗2] ∪ · · · ∪ ]p

∗r ,+∞[ with p∗1, . . . , p∗r nonzero singular values of loading.

Forgetting the p-dependency means that p /∈ {0, p∗1, . . . , p∗r }. In the examplesexcepted for Bigoni’s system, 0 is the only singular value. The singular problemis not investigated in this paper.

Let Fa = Im(φa) and Ga = Ker(φa). We know that (as every skew-symmetriclinear map) φa has an even rank, r = 2`, and that its kernel and its image areorthogonal spaces. Let Ga = Ker(φa). Thus Rn

= Fa⊥

⊕Ga . We set the following:

Definition. The integer ` is called the geometric degree of nonconservativity of6free.

As φ2a is a symmetric linear mapping it is diagonalizable in an orthonormal basis.

Moreover Ga = Ker(φa) = Ker(φ2a), the nonzero eigenvalues of φ2

a are negative,and the associated eigenspaces are two-dimensional and mutually orthogonal. Notethese values −µ2

1, . . . ,−µ2` and E

−µ2i, the associated eigenspaces for i = 1, . . . , `.

Each of these spaces are φa-stable. Because of the φa-stability of each of the spacesof the decomposition

Rn= Ga

⊥

⊕ E−µ2

1

⊥

⊕ . . .⊥

⊕ E−µ2

`,

we deduce (by Cartan’s theorem) the existence of an orthonormal basis b′ of Rn

128 LERBET, ALDOWAJI, CHALLAMEL, KIRILLOV, NICOT AND DARVE

such that the matrix of φa in b′ is

0 · · · · · · 0.... . .

...

0 · · ·

0 −µ1

µ1 0. . .

... 0 −µ`0 µ` 0

.

Thus

Rn= Ga

⊥

⊕ E−µ2

1

⊥

⊕ · · ·⊥

⊕ E−µ2

`= Fa

⊥

⊕Ga = H(A)⊥

⊕ T (A).

Proposition. Equation (7) holds if and only if m= ` and the family A of constraintsmust be built by choosing αi

∈ E−µ2

ifor i = 1, . . . , `= m, being, however, careful

that the property fails if two constraints are chosen in the same eigenspace E−µ2i.

Proof. Suppose first that m = ` = 12 rank(φa) and A built as proposed in the

proposition. Let u ∈ H(A). Complete the basis (α1, . . . , αm) by n− 2m vectorsβ1, . . . , βn−2m of Ga and the m other vectors (φa(α

1), . . . , φa(αm)) so that the

family (β1, . . . βn−2m, φa(α1), . . . , φa(α

m)) is an orthogonal basis of T (A), as maybe easily checked. By definition,

u =m∑

k=1

ukαk,

and thenφa(p)(u)=

m∑k=1

ukφ(αk) ∈ T (A),

which is exactly (7).Reciprocally, suppose now m < l or m = l but A is not built as proposed in the

proposition. Thus there is some i ∈ {1, . . . , l} such that any α j belongs to E−µ2

imeaning geometrically that T (A)∩E

−µ2i={0}. Choose now u 6= 0 in H(A)∩E

−µ2i.

Thus (u, φa(u)) is an orthogonal basis of E−µ2

i, meaning that φa(u)(6= 0) ∈ E

−µ2i,

implying φa(u) /∈ T (A). �

Thus, coming back to the mechanical problem, a free nonconservative mechani-cal system 6free can be made conservative by means of m constraints if and only ifKa(p) has rank 2m and the matrix A is formed by by m vectors α1, . . . , αm , eachαi being chosen in the eigenspace E

−µ2i

of Ka(p)2; and Cm(6)= {A ∈Mnm(R) |

coli (A) ∈ E−µ2

i\ {0}} (with obvious notations) is an open 2m-dimensional cone of

Mnm(R).

GEOMETRIC DEGREE OF NONCONSERVATIVITY 129

As stated previously, any constrained conservative system is still a conservativesystem. Thus if there are k ≥m constraints, and m of the k constraints are chosen asabove, the constrained system is still conservative. If rank(Ka(p))= 2m, then m isthe minimum number of constraints needed to convert the system into a conserva-tive one. The nonconservativity of the free system 6free is then characterized by twomeasures of nonconservativity. The first is the norm of the skew-symmetric partKa(p), which indicates the amplitude of the nonconservativity, while the second isthe rank 2` of Ka(p), which acts as a geometric measure of the nonconservativityor a sort of dimension (`) of the nonconservativity. This is the reason for theabove definition of the geometric degree of nonconservativity of 6free. Moreoverwe may localize this nonconservativity because we may build families of ` vectors(or constraints) allowing us to convert the initial nonconservative system into aconservative one. Note also that the proof is constructive because it builds the kine-matic constraints A converting the system 6free into a conservative one (6cons(A)).There are 2m different independent systems A of constraints converting the non-conservative 6free into a conservative 6cons(A). This result may be consideredas a sort of dual to the result about the destabilizing effect of adding kinematicconstraints in nonconservative systems (see again [Challamel et al. 2009; 2010;Nicot et al. 2011; Lerbet et al. 2012]): by adding a suitable constraint in a suitableeigenspace of Ks , one can destabilize a stable nonconservative system. Here, bychoosing appropriate constraints in suitable eigenspaces of K 2

a , one can convert anonconservative system into a conservative one. In the following section, severalexamples issued from different mechanical systems illustrate these results.

3. Examples

In this section, we propose a collection of examples consisting in variations on theparadigmatic Ziegler column. The degree of freedom (parameter n) and the natureof the follower force (partial or complete follower force parameter γ ) may change.In the most general case, the system 6 consists of n bars O A1, A1 A2, . . . , An−1 An

with O A1 = A1 A2 = · · · = An−1 An = h linked with n elastic springs with the samestiffness k. EP is the follower nonconservative load acting on An . Adopting adimensionless format, we use p = ‖ EP‖h/k as a loading parameter. To investigatehow the algebraic method is performing, we conduct the complete calculation onlyfor the three-DOF Ziegler column.

In Section 3.1, we investigate the pure Ziegler system and we notice that thegeometric degree of nonconservativity is one for any number of rigid bars, meaningfor any degree of freedom. Increasing the number of bars or the degree of freedomdoes not change its geometric degree of nonconservativity: from the geometricpoint of view, the Ziegler system is weakly nonconservative. In Section 3.2, we

130 LERBET, ALDOWAJI, CHALLAMEL, KIRILLOV, NICOT AND DARVE

investigate what we call the multiple-DOF Bigoni system, because it involves de-vice like that of [Bigoni and Noselli 2011] at each joint. This system appears asa generalization of the n-DOF Ziegler column where the load parameter is itselfdistributed on the system and may vary on each joint. It also may be considered asa discretized Leipholz column [Leipholz 1987]. In this case, the geometric degreeof nonconservativity increases with the number of bars and the degree of freedom.Calculations are made only for n = 2 and n = 4. From a geometric point of view,the Bigoni system is essentially more strongly nonconservative than the Zieglersystem.

3.1. Ziegler systems.

3.1.1. Two-DOF Ziegler column with complete follower force. The geometric stiff-ness matrix is

Kext =

(1 −10 0

).

Its skew-symmetric part is

Ka,ext =

(0 − 1

212 0

).

The square of Ka,ext is

K 2a,ext =−

14

(1 00 1

),

where µ21 = −

14 , K 2

a,ext is spheric, and Eµ1(φ2a,ext) = R2. Then α is any vector in

R2. Obviously any constraint converts the free system into a conservative one as aone-DOF (elastic) system is always conservative because any continuous functionhas a primitive.

3.1.2. Two-DOF Ziegler column with partial follower force. The geometric stiff-ness matrix is

Kext =

(1 −γ0 1−γ

).

Its skew-symmetric part is

Ka,ext =

0 −γ2

γ

20

.The square of Ka,ext is

K 2a,ext =−

γ 2

4

(1 00 1

).

Our conclusions are similar to those above.

GEOMETRIC DEGREE OF NONCONSERVATIVITY 131

3.1.3. Three-DOF Ziegler column with complete follower force. The geometricstiffness matrix is

Kext =

1 0 −10 1 −10 0 0

.Its skew-symmetric part is

Ka,ext =12

0 0 −10 0 −11 1 0

.Obviously rank(Ka,ext)= 2. The square of Ka,ext is:

K 2a,ext =−

14

1 1 01 1 00 0 2

.Calculations give −µ2

1 =−12 and

E−

12(K 2

a,ext)= Vec

α =0

01

, Ka,extα =12

−1−1

0

,leading to two generic constraints converting the system into a conservative one:θ3 = 0 and θ1+ θ2 = 0. In practice, any linear combination of these two constraintslies in the corresponding plane and may be chosen as a possible constraint convert-ing the system into a conservative one. We now propose to check for this case theresults coming from our algebraic method with respect to the direct approach tothe problem.

The virtual power of the follower force reads:

P∗(P)= Q1θ∗

1 + Q2θ∗

2 + Q3θ∗

3 = Ph(sin(θ3− θ1)θ∗

1 + sin(θ3− θ2)θ∗

2 ). (8)

The complete nonlinear condition in order to have a conservative system is thatthere is a function θ= (θ1, θ2, θ3) 7→U (θ1, θ2, θ3)=U (θ) such that Qk=−∂U/∂θk ,which is here obviously impossible without an additional constraint: that the freesystem is nonconservative!

Suppose now the system is subjected to a kinematic constraint φ(θ)= 0, whichleads to the following condition on the virtual parameters:

∂φ

∂θ1θ∗1 +

∂φ

∂θ2θ∗2 +

∂φ

∂θ3θ∗3 = 0. (9)

Supposing the problem is resolvable with respect to the variable θ3, meaning that∂φ/∂θ3 6= 0, we deduce from the implicit functions theorem that (locally in the

132 LERBET, ALDOWAJI, CHALLAMEL, KIRILLOV, NICOT AND DARVE

neighborhood of θ = 0), θ3 = θ3(θ1, θ2) meaning that, to first order,

θ3 = θ3(θ1, θ2)≈∂θ3

∂θ1

∣∣∣∣θ=0θ1+

∂θ3

∂θ2

∣∣∣∣θ=0θ2 = c1θ1+ c2θ2. (10)

Thus, to first order, the expansion reads: Q1 = Q1(θ1, θ2)≈ Ph((c1−1)θ1+ c2θ2)

and Q2 = Q2(θ1, θ2)≈ Ph((c1θ1+ (c2− 1)θ2). The condition of conservativity ofthe loading then reads c1 = c2 = c. The kinematic relation is θ3 = c(θ1+ θ2) andthe quadratic potential is

U (θ)≈−Ph(c−1

2(θ2

1 + θ22 )+ cθ1θ2

). (11)

For c = 0, we find the first generic kinematic constraint θ3 = 0, and for c 6= 0 it is,as expected, a linear combination of both generic constraints.

Suppose now that the problem is not resolvable with respect to the variable θ3,meaning that ∂φ/∂θ3 = 0. We then deduce that, linearly, the relation only concernsθ1 and θ2 and reads linearly as

∂φ

∂θ1

∣∣∣∣θ=0

θ1+∂φ

∂θ2

∣∣∣∣θ=0

θ2 = a1θ1+ a2θ2 ≈ 0, (12)

and that a1θ∗

1 + a2θ∗

2 = 0. Resolving these relations, for example, with respect tothe variable θ1 (θ1 =−bθ2 =−(a2/a1)θ2 and reporting this relation in (8) showsthat P∗(P) = Q2θ

∗

2 with Q2 = Q2(θ2, θ3) ≈ Ph(−b(θ3 − bθ2) + (θ3 − θ2)) ≈

Ph((b2− 1)θ2+ (1− b)θ3). The condition of integrability then reads b = 1, the

kinematic relation is θ1+ θ2 = 0, and the potential is nil up to order two. Then wewill come back precisely to the second generic kinematic constraint. To sum up,the direct calculations lead to both generic constraints obtained from our algebraicmethod.

3.1.4. Three-DOF Ziegler column with partial follower force. The geometric stiff-ness matrix is

Kext =

1 0 −γ0 1 −γ0 0 1− γ

.Its skew-symmetric part is

Ka,ext =12γ

0 0 −10 0 −11 1 0

.

GEOMETRIC DEGREE OF NONCONSERVATIVITY 133

Obviously rank(Ka,ext)= 2. The square of Ka,ext is

K 2a,ext =−

14γ 2

1 1 01 1 00 0 2

.Calculations give −µ2

1 =−γ2/2 and

E−γ 2/2(K2a,ext)= Vec

α =0

01

, Ka,extα =γ

2

−1−1

0

leading to the same two generic constraints as previously which convert the systeminto a conservative one: θ3 = 0 and θ1+ θ2 = 0.

3.1.5. An n-DOF Ziegler column with complete follower force (γ = 1). The stiff-ness matrix is

K (p)=

2−p −1 0 0 · · · 0 p−1 2−p −1 0 · · · 0 p0 −1 2−p −1 · · · 0 p...

......

.... . .

......

0 0 0 0 · · · 2−p −1+p0 0 0 0 · · · −1 1

.

Its skew-symmetric part is

Ka(p)=p2

0 0 0 0 · · · 0 10 0 0 0 · · · 0 10 0 0 0 · · · 0 1...

......

.... . .

......

0 0 0 0 · · · 0 1−1 −1 −1 −1 · · · −1 0

.

Obviously rank(Ka(p))= 2. The square of Ka(p) is

K 2a (p)=−

p2

4

1 1 1 1 · · · 1 01 1 1 1 · · · 1 01 1 1 1 · · · 1 0............. . .

......

1 1 1 1 · · · 1 00 0 0 0 · · · 0 n−1

= p2 K 2

a .

134 LERBET, ALDOWAJI, CHALLAMEL, KIRILLOV, NICOT AND DARVE

X

Yp

Ok

k

k

k

k

k A

ϑ

n

n

ϑ4

ϑ3

ϑ2

ϑ1

An

−1

An−2

An−3

A4

A3

A2

A1

ϑn−1

X

Y p

Ok

k

k

k

k

k

ϑ n

ϑ4

ϑ3

ϑ2

ϑ1

An

An−1

An−2

= 0

A3

A2

A1

ϑn−1

Figure 1. An n-DOF Ziegler column with complete followerforce: the case θ1 + · · · + θn−1 = 0 (left) and the case θn = 0(right).

Calculations give −µ21 =−(n− 1)/4 and

E−(n−1)/4(K 2a )= Vec

α =

000...

01

, K 2aα =

12

−1−1−1...

−10

leading to two generic constraints converting the system into a conservative one:θ1+ · · ·+ θn−1 = 0, meaning that the motion of An−1 is constrained to remain onthe axis OY (Figure 1, left), and θn = 0 (Figure 1, right).

3.2. The Bigoni system or discretized Leipholz column. We now turn to the n-DOF Bigoni system [Bigoni and Noselli 2011], which can also be regarded as ann-DOF Leipholz column [1987]. The system 6 consists of n bars O A1, A1 A2,

GEOMETRIC DEGREE OF NONCONSERVATIVITY 135

. . . , An−1 An , with O A1 = A1 A2 = · · · = An−1 An = h linked with n elastic springswith the same stiffness k. Adopting the same device at the end of each bar of6 leads to a family of follower forces EP1, . . . , EPn (see Figure 2, left). The purefollower forces EP1, EP2, . . . , EPn are applied at the ends of O A1, A1 A2, . . . , An−1 An ,respectively. Adopting a dimensionless format, we use pi = ‖ EPi‖h/k, for i =1, . . . , n, as loading parameters. The stiffness matrix is K (p)= K (p1, p2, . . . , pn):

K (p)=

2−n∑

i=2

pi −1+p2 p3 p4 p5 · · · pn−1 pn

−1 2−n∑

i=3

pi −1+p3 p4 p5 · · · pn−1 pn

0 −1 2−n∑

i=4

pi −1+p4 p5 · · · pn−1 pn

0 0 −1 2−n∑

i=5

pi −1+p5 · · · pn−1 pn

......

......

.... . .

......

0 0 0 0 0 · · · −1+pn−1 pn

0 0 0 0 0 · · · 2−pn −1+pn

0 0 0 0 0 · · · −1 1

.

Its skew-symmetric part is

Ka(p)=12

0 p2 p3 p4 p5 · · · pn−1 pn

−p2 0 p3 p4 p5 · · · pn−1 pn

−p3 −p3 0 p4 p5 · · · pn−1 pn

−p4 −p4 −p4 0 p5 · · · pn−1 pn...

......

......

. . ....

...

−pn−1 −pn−1 −pn−1 −pn−1 −pn−1 · · · 0 pn

−pn −pn −pn −pn −pn · · · −pn 0

and

rank(Ka(p))=

{n if n even,

n− 1 if n odd,

thus

`=

{n/2 if n even,

(n− 1)/2 if n odd.

For n = 2, calculations give

Ka2=−

14

p22

(1 00 1

), −µ2

1 =−14

p22.

136 LERBET, ALDOWAJI, CHALLAMEL, KIRILLOV, NICOT AND DARVE

X

Y

Ok

k

k

k

k

k

ϑ n

ϑ4

ϑ3

ϑ2

ϑ1

An

An−1

pn−1

pn

An−2

A3

A2

A1

ϑn−1

p3

p1

p2

X

Y

Ok

k-1

A2

A1

p1

p2

-�2= 0

X X

Y

Ok

k

-2

A2

A1

p1

p2

-�1= 0

Figure 2. Bigoni systems with n DOF (left) and two DOF (right).

K 2a is spherical, E

−µ21(K 2

a ) = R2, and α is then any vector in R2. The geometricdegree of nonconservativity is equal to 1 and the constraint is a linear combinationof the two generic constraints θ1 = 0 and θ2 = 0: this is any linear constraint! (SeeFigure 2, right.)

For n = 4, calculations give

−µ21 =−

38 p2

4 −14 p2

3 −18 p2

2 +18 a, −µ2

2 =−38 p2

4 −14 p2

3 −18 p2

2 −18 a,

where a =√

9p44 + 12p2

3 p24 + 2p2

4 p22 + 4p4

3 + 4p23 p2

2 + p42 , and

E−µ2

1(K 2

a )= Vec{α1 =

[[2(−p2

3 p24a+ 2p3 p2 p2

4a− p24 p2

2a+ p3 p32a

+ p33 p2a− p2

4 p42 + 3p3 p3

2 p24 + 5p3

3 p2 p24 + 2p3 p2 p4

4 + p23 p2

2 p24 + p4

4 p22

+ 7p44 p2

3 + 6p64 + 2p2

4 p43 − 2p4

4a+ 2p53 p2+ 3p3

3 p32 + p3 p5

2)],

[−(2p33 p2+ 5p3 p2 p2

4 + p3 p32 + 3p2

4 p22 + p3 p2a+ 3p4

4

+ 2p23 p2

4 − p24a)(−p2

4 + p22 + a)], [(−3p4

4 − 2p23 p2

4 + 2p24 p2

2

+ p24a+ 2p2

3 p22 + p4

2 + p22a)(−p4

2+ p2

2+ a)], [0]

]}, (13)

GEOMETRIC DEGREE OF NONCONSERVATIVITY 137

E−µ2

2(K 2

a )= Vec{α2 =

[[2p4(p3

3a+ p32a+ p2

3 p2a+ p3 p22a

+ 2p24 p2a+ 2p2

4 p3a+ p3 p42 + 2p4

4 p2+ 3p24 p3

2 + 3p23 p3

2 − 7p33 p2

4

− 6p3 p44 − 2p5

3 − p24 p2

2 p3+ 5p24 p2

3 p2− p22 p3

3 + 2p43 p2+ p5

2)],

[−p4(2p23 p2+ 5p2

4 p2+ p32 − 3p3 p2

2 + p2a− 3p24 p3− 2p3

3

+ p3a)(−p24 + p2

2 + a)], [0], [(−3p44 − 2p2

3 p24

+ 2p24 p2

2 + p24a+ 2p2

3 p22 + p4

2 + p22a)(−p2

4 + p22 + a)]

]}. (14)

For pi =cih

, we have

−µ21 =

c2

1152h2

(−95+

√7729

),

−µ22 =

c2

1152h2

(−95−

√7729

),

a = 1144

√7729c2

h2 ,

so

E−µ2

1(K 2

a )= Vec

α1 =

c6

h6

356

√7729+137

27+√

7729

−1

24

(281−

√7729

)18

(57+√

7729)

0

, Kaα1

,

E−µ2

2(K 2

a )= Vec

α2 =c6

h6

353

√7729+37

27+√

7729

−1

12

(1−√

7729)

018

(57+√

7729)

, Kaα2

.In this example, the geometric degree of nonconservativity is equal to 2: two ad-ditional kinematic constraints φ1(θ1, . . . , θ4)= 0 and φ2(θ1, . . . , θ4)= 0 are thennecessary to convert the system into a conservative one, each constraint φi beingchosen in E

−µ2i(K 2

a ) for i = 1, 2. For example,

φ1(θ1, . . . , θ4)=356

√7729+137

27+√

7729θ1−

124

(281−

√7729

)θ2+

18

(57+√

7729)θ3,

φ2(θ1, . . . , θ4)=356

√7729+137

27+√

7729θ1−

112

(1−√

7729)θ2+

18

(57+√

7729)θ4.

Conclusion

In this paper, we investigate nonconservative systems, meaning here elastic systemswith a nonsymmetric stiffness matrix. We associate with each mechanical system

138 LERBET, ALDOWAJI, CHALLAMEL, KIRILLOV, NICOT AND DARVE

a minimal number ` of additional kinematic constraints allowing this system to beconverted into a conservative one. As this integer measure of the nonconservativityof the mechanical system is linked with the dimension of a vector space, it is calledthe geometric degree of nonconservativity of the system. Computations of thisinteger and of the corresponding additional kinematic constraints are constructiveand several examples illustrate the results. The extension to the nonlinear case willbe developed in a forthcoming paper.

References

[Absi and Lerbet 2004] E. Absi and J. Lerbet, “Instability of elastic bodies”, Mech. Res. Commun.31 (2004), 39–44.

[Bigoni and Noselli 2011] D. Bigoni and G. Noselli, “Experimental evidence of flutter and diver-gence instabilities induced by dry friction”, J. Mech. Phys. Solids 59 (2011), 2208–2226.

[Bolotin 1961] V. V. Bolotin, Nekonservativnye zadaqi teorii uprugoi ustoiqivosti,Fizmatlit, Moscow, 1961. Translated as Nonconservative problems of the theory of elastic stability,Macmillan, New York, 1963.

[Challamel et al. 2009] N. Challamel, F. Nicot, J. Lerbet, and F. Darve, “On the stability of non-conservative elastic systems under mixed perturbations”, Eur. J. Environ. Civ. Eng. 13:3 (2009),347–367.

[Challamel et al. 2010] N. Challamel, F. Nicot, J. Lerbet, and F. Darve, “Stability of non-conservativeelastic structures under additional kinematic constraints”, Eng. Struct. 32:10 (2010), 3086–3092.

[Hill 1958] R. Hill, “A general theory of uniqueness and stability in elastic-plastic solids”, J. Mech.Phys. Solids 6:3 (1958), 236–249.

[Kirillov and Verhulst 2010] O. N. Kirillov and F. Verhulst, “Paradoxes of dissipation-induced desta-bilization or who opened Whitney’s umbrella?”, Z. Angew. Math. Mech. 90:6 (2010), 462–488.

[Leipholz 1987] H. Leipholz, Stability theory: an introduction to the stability of dynamic systemsand rigid bodies, 2nd ed., Wiley, New York, 1987.

[Lerbet et al. 2012] J. Lerbet, M. Aldowaji, N. Challamel, F. Nicot, F. Prunier, and F. Darve, “P-positive definite matrices and stability of nonconservative systems”, Z. Angew. Math. Mech. 92:5(2012), 409–422.

[Lerbet et al. 2013] J. Lerbet, O. Kirillov, M. Aldowaji, N. Challamel, F. Nicot, and F. Darve, “Addi-tional constraints may soften a nonconservative structural system: buckling and vibration analysis”,Int. J. Solids Struct. 50:2 (2013), 363–370.

[Nicot et al. 2011] F. Nicot, N. Challamel, J. Lerbet, F. Prunier, and F. Darve, “Bifurcation andgeneralized mixed loading conditions in geomaterials”, Int. J. Numer. Anal. Methods Geomech.35:13 (2011), 1409–1431.

[Zhuravlëv 2007] V. F. Zhuravlëv, “O razlo�enii nelineinyh obobwennyh sil na po-tencial~nu� i cirkul�rnu� komponenty”, Dokl. Phys. 414:5 (2007), 622–624. Translatedas “Decomposition of nonlinear generalized forces into potential and circulatory components” inDoklady Physics 52:6 (2007), 339–341.

[Ziegler 1952] H. Ziegler, “Die Stabilitätskriterien der Elastomechanik”, Ing. Arch. 20 (1952), 49–56.

Received 9 Nov 2012. Revised 15 Apr 2013. Accepted 25 May 2013.

GEOMETRIC DEGREE OF NONCONSERVATIVITY 139

JEAN LERBET: jean.lerbet@ibisc.univ-evry.frInformatique Biologie Intégrative Systèmes Complexes, Université d’Evry-Val-d’Essonne,UFR Sciences & Technologies, 40, Rue du Pelvoux, CE 1455 Courcouronnes, 91020 Evry cedex,France

MARWA ALDOWAJI: marwadoh@yahoo.comInformatique Biologie Intégrative Systèmes Complexes, Université d’Evry-Val-d’Essonne,UFR Sciences & Technologies, 40, Rue du Pelvoux, CE 1455 Courcouronnes, 91020 Evry cedex,France

NOËL CHALLAMEL: noel.challamel@univ-ubs.frLaboratoire d’Ingénierie des Matériaux de Bretagne, Université de Bretagne Sud,Rue de saint Maudé, BP 92116, 56321 Lorient cedex, France

OLEG N. KIRILLOV: o.kirillov@hzdr.deDepartment of Magnetohydrodynamics, Institute of Fluid Dynamics, Helmholtz-ZentrumDresden-Rossendorf, Bautzner Landstraße 400, P.O. Box 510119, 01314 Dresden, Germany

FRANÇOIS NICOT: Francois.Nicot@irstea.frGeomechanics Group, Irstea, Domaine Universitaire BP 76, 38402 Saint Martin d’Hères cedex,France

FÉLIX DARVE: felix.darve@3sr-grenoble.frLaboratoire Sols Solides Structures, UJF-INPG-CNRS, BP 53, 38041 Grenoble cedex 9, France

MM ∩msp

MATHEMATICS AND MECHANICS OF COMPLEX SYSTEMSmsp.org/memocs

EDITORIAL BOARDANTONIO CARCATERRA Università di Roma “La Sapienza”, Italia

ERIC A. CARLEN Rutgers University, USAFRANCESCO DELL’ISOLA (CO-CHAIR) Università di Roma “La Sapienza”, Italia

RAFFAELE ESPOSITO (TREASURER) Università dell’Aquila, ItaliaALBERT FANNJIANG University of California at Davis, USA

GILLES A. FRANCFORT (CO-CHAIR) Université Paris-Nord, FrancePIERANGELO MARCATI Università dell’Aquila, Italy

JEAN-JACQUES MARIGO École Polytechnique, FrancePETER A. MARKOWICH DAMTP Cambridge, UK, and University of Vienna, Austria

MARTIN OSTOJA-STARZEWSKI (CHAIR MANAGING EDITOR) Univ. of Illinois at Urbana-Champaign, USAPIERRE SEPPECHER Université du Sud Toulon-Var, France

DAVID J. STEIGMANN University of California at Berkeley, USAPAUL STEINMANN Universität Erlangen-Nürnberg, Germany

PIERRE M. SUQUET LMA CNRS Marseille, France

MANAGING EDITORSMICOL AMAR Università di Roma “La Sapienza”, Italia

CORRADO LATTANZIO Università dell’Aquila, ItalyANGELA MADEO Université de Lyon–INSA (Institut National des Sciences Appliquées), France

MARTIN OSTOJA-STARZEWSKI (CHAIR MANAGING EDITOR) Univ. of Illinois at Urbana-Champaign, USA

ADVISORY BOARDADNAN AKAY Carnegie Mellon University, USA, and Bilkent University, Turkey

HOLM ALTENBACH Otto-von-Guericke-Universität Magdeburg, GermanyMICOL AMAR Università di Roma “La Sapienza”, ItaliaHARM ASKES University of Sheffield, UK

TEODOR ATANACKOVIC University of Novi Sad, SerbiaVICTOR BERDICHEVSKY Wayne State University, USA

GUY BOUCHITTÉ Université du Sud Toulon-Var, FranceANDREA BRAIDES Università di Roma Tor Vergata, Italia

ROBERTO CAMASSA University of North Carolina at Chapel Hill, USAMAURO CARFORE Università di Pavia, Italia

ERIC DARVE Stanford University, USAFELIX DARVE Institut Polytechnique de Grenoble, France

ANNA DE MASI Università dell’Aquila, ItaliaGIANPIETRO DEL PIERO Università di Ferrara and International Research Center MEMOCS, Italia

EMMANUELE DI BENEDETTO Vanderbilt University, USABERNOLD FIEDLER Freie Universität Berlin, Germany

IRENE M. GAMBA University of Texas at Austin, USASERGEY GAVRILYUK Université Aix-Marseille, FranceTIMOTHY J. HEALEY Cornell University, USADOMINIQUE JEULIN École des Mines, FranceROGER E. KHAYAT University of Western Ontario, Canada

CORRADO LATTANZIO Università dell’Aquila, ItalyROBERT P. LIPTON Louisiana State University, USAANGELO LUONGO Università dell’Aquila, ItaliaANGELA MADEO Université de Lyon–INSA (Institut National des Sciences Appliquées), France

JUAN J. MANFREDI University of Pittsburgh, USACARLO MARCHIORO Università di Roma “La Sapienza”, ItaliaGÉRARD A. MAUGIN Université Paris VI, FranceROBERTO NATALINI Istituto per le Applicazioni del Calcolo “M. Picone”, Italy

PATRIZIO NEFF Universität Duisburg-Essen, GermanyANDREY PIATNITSKI Narvik University College, Norway, Russia

ERRICO PRESUTTI Università di Roma Tor Vergata, ItalyMARIO PULVIRENTI Università di Roma “La Sapienza”, Italia

LUCIO RUSSO Università di Roma “Tor Vergata”, ItaliaMIGUEL A. F. SANJUAN Universidad Rey Juan Carlos, Madrid, Spain

PATRICK SELVADURAI McGill University, CanadaALEXANDER P. SEYRANIAN Moscow State Lomonosov University, Russia

MIROSLAV ŠILHAVÝ Academy of Sciences of the Czech RepublicGUIDO SWEERS Universität zu Köln, Germany

ANTOINETTE TORDESILLAS University of Melbourne, AustraliaLEV TRUSKINOVSKY École Polytechnique, France

JUAN J. L. VELÁZQUEZ Bonn University, GermanyVINCENZO VESPRI Università di Firenze, ItaliaANGELO VULPIANI Università di Roma La Sapienza, Italia

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109A mixed boundary value problem in potential theory for abimaterial porous region: An application in the environmentalgeosciences

A. P. S. Selvadurai

123Geometric degree of nonconservativityJean Lerbet, Marwa Aldowaji, Noël Challamel, Oleg N.Kirillov, François Nicot and Félix Darve

141Asymptotic analysis of small defects near a singular point inantiplane elasticity, with an application to the nucleation of a crackat a notch

Thi Bach Tuyet Dang, Laurence Halpern and Jean-JacquesMarigo

181The homogenized behavior of unidirectional fiber-reinforcedcomposite materials in the case of debonded fibers

Yahya Berrehili and Jean-Jacques Marigo

209Statistically isotropic tensor random fields: Correlation structuresAnatoliy Malyarenko and Martin Ostoja-Starzewski

MEMOCS is a journal of the International Research Center forthe Mathematics and Mechanics of Complex Systemsat the Università dell’Aquila, Italy.

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